Basics Skills: Capstone
Table of Contents:
(1.) Inferential Statistics: Introduction
(2.) Inferential Statistics: An Example
(3.) Using Inferential Statistics
(4.) The t-test
(5.)The t-test - An Example
(6.) t-tests and Statistical Assumptions - A Word of Caution
(7.) Answers to Practice Problems
(8.)Sample Problems
(9.)Capstone: Putting It All Together
Inferential Statistics: Introduction
You have already gained experience with descriptive statistics but we will now introduce a new class of statistics that are also useful in science - inferential statistics. Before I define inferential statistics, let me show you why they are useful. In the previous section, I described a research study that sought to determine the effects of temperature on plant biomass. Upon completion of her experiment, the experimenter would have sets of biomass measurements for each group. What does she do then? Should she take the mean of the values for each group and then make conclusions based on this statistic alone? What if the mean biomass for one group is only slightly higher than that for another group - is the difference sufficient for her to make a solid conclusion? Inferential statistics allow you to make comparisons in scientific studies and determine with confidence if differences in treatment groups truly exist.
Inferential Statistics: An Example
Inferential statistics are used to make comparisons between data sets and infer whether the two data sets are significantly different from one another. It is important to realize that when dealing with statistics and probability, chance always plays a role. When we compare means from two groups in an experiment, we are attempting to determine if the two means truly differ from one another, or if the difference in the means of the groups is simply due to random chance. The best way to explain this concept is with an example.
After losing a close game in overtime, a local high school football coach accuses the officials of using a "loaded" coin during the pre-overtime coin toss. He claims that the coin was altered to come up heads when flipped, his opponents knew this, won the coin toss, and consequently won the game on their first possession in overtime. He wants the local high school athletic association to investigate the matter. You are assigned the task of determining if /
the coach's accusation stands up to scrutiny. Well, you know that a "fair" coin should land on heads 50% of the time, and on tails 50% of the time. So how can you test if the coin in question is doctored? If you flip it ten times and it comes up heads six times, does that validate the accusation? What if it comes up heads seven times? What about eight times? To make a conclusion, you need to know the probability of these occurrences.
To examine the potential outcomes of coin flipping, we will use a Binomial Distribution. This distribution describes the probabilities for events when you have two possible outcomes (heads or tails) and independent trials (one flip of the coin does not influence the next flip). The distribution for ten flips of a fair coin is shown in Figure 1.
Figure 1. Binomial distribution for fair coin with ten flips
Note that ratio of 5 heads:5 tails is the most probable, and the probabilities of other combinations decline as you approach greater numbers of heads or tails. The figure demonstrates two important points. One, it shows that the expected outcome is the most probable - in this case a 5:5 ratio of heads to tails. Two, it shows that unlikely events can happen due solely to random chance (e.g., getting 0 heads and 10 tails), but that they have a very low probability of occurring.
Also note that the binomial distribution is rather "jagged" when only ten coin flips are performed. As the number of trials (coin flips) increases, the shape of the distribution begins to smooth out and resemble a normal curve. Note how the shape of the curve with 50 trials is much smoother than the curve for 10 trials, and more representative of a normal curve (Figure 2). /Figure 2. Binomial distribution for fair coin with 50 flips
Inferential Statistics: Probability
Normal curves are useful because they allow us to make statistical conclusions about the likelihood of being a certain distance from the center (mean) of the distribution. In a normal distribution, there are probabilities associated with differing distances from the mean. Recall that 68% of the values in a distribution are within one standard deviation of the mean, 95% of values are within two standard deviations of the mean, and 99% of the values are within three standard deviations of the mean.
Figure 3. Probabilities for results of 10 coin flips [ animate ]
The difficulty with working with probabilities is knowing when to conclude that an occurrence is not due to random chance. Values far from the mean in a distribution can occur, but will occur with low probability (Figure 3). We are therefore essentially testing the hypothesis that the observed data fits a particular distribution. In the coin flip example, we're testing to see if our results fit those expected from the distribution of a fair coin. So we need to come up with a point at which we can conclude our results are definitely not part of the distribution we are testing. So when do you determine that a given data set no longer fits a distribution when random chance will always play a role? Well, you've got to make an arbitrary decision, and statisticians set precedent long ago. Given that 95% of the values in a distribution fall within two standard deviations of the mean, statisticians have decided that if a result falls outside of this range, you can determine that your data does not fit the distribution you are testing. This essentially says that if your result has equal to or less than a 5% chance of belonging to a particular distribution, then you can conclude it is not a part of that distribution. As probabilities are listed as proportions, this means that a result is "statistically significant" if its occurrence is equal to or less than 0.05. This leads to our statistical "rule of thumb" - whenever a statistical test returns a probability value (or "p-value") equal to or less than 0.05, we reject the hypothesis that our results fit the distribution we are testing. The standard practice in such comparisons is to use a null hypothesis (written as "Ho"), which states that the data fits the distribution.
Ho: The data fit the assigned distribution
To practice your interpretation of p-values, decide if each of the p-values below indicates that you should reject your null hypothesis. Answers are provided at the end of the exercise.
PRACTICE PROBLEM #1
p = 0.11Reject or Do Not Reject Ho?
p = 0.56Reject or Do Not Reject Ho?
p = 0.99Reject or Do Not Reject Ho?
p = 0.01Reject or Do Not Reject Ho?
p < 0.005Reject or Do Not Reject Ho?
So our coin test is comparing our result to the distribution of a fair coin. To test the coin, you opt to flip it 50 times, tally the number of heads and tails, and compare your results to the fair coin distribution. You obtain the results listed below.
Heads / 33Tails / 17
So what does this mean? Referencing the distribution (Figure 2 below), we see that a ratio of 33 heads to 17 tails would only occur about 1% of the time if the coin were indeed fair. As this is less than 5% (p < 0.05), we can reject our hypothesis that the result fits the distribution. We were testing the distribution of a fair coin, so this suggests the coin was not fair, and the coach's accusation has merit. If we look at a distribution for a rigged coin that comes up heads 70% of the time instead of 50% of the time (Figure 4), we notice that our result fits quite well into this distribution. This indicates that further tests should be conducted, and the number of trials (coin flips) increased so a more definitive conclusion could be reached. Man, I love a good controversy...
Figure 2 (again). Binomial distribution for fair coin with 50 flips
Figure 4. Binomial distribution for coin rigged towards "heads"
Using Inferential Statistics
In the coin flip example, we tested to see if the results of our tests fit the distribution of a fair coin based on the predicted probabilities of the various outcomes. Most statistical tests in the sciences do not work this way, however. In most cases, scientists manipulate variables in experiments, and then make comparisons between groups receiving different treatments. For example, if you are investigating the effects of different brands of fertilizer on tomato plant growth, you would be interested in comparing the growth of the plants in the different treatments to one another. In a situation like this, we are interested in comparing two groups to one another, rather than comparing one group to an existing distribution. To do this, we will use a statistical test that compares the distributions of two data sets to one another - the t-test.
The t-test
The t-test is an inferential statistic that enables you to compare the means of two groups and determine if they are statistically different from one another. In essence, the test compares the distributions of the two data sets to one another, and tests the hypothesis that the two data sets belong to the same distribution. If there is a low likelihood that the two data sets belong to the same distribution (probability less than or equal to 5%), then we can conclude that true differences in the means of the two groups exist, and the two groups are significantly different from one another. The t-test accomplishes this task by looking at both the mean and the dispersion of the data in the two groups. Figures 5 and 6 will help to illustrate how a t-test works:
Let's say we've got data from two groups that we wish to compare. To help visualize things, we can graph the distributions of the two data sets on one graph so we can see the mean and dispersion for the two groups (Figure 5).
Figure 5. The t-test /
Figure 6. t-test comparison of distributions
A t-test compares both the means and distributions of data sets in order to determine if they are different from one another. Why does it do this? Why not just look at the means and make conclusions based on that? As illustrated here, the dispersion of the data tells you a great deal about the data set. Note that in each of the three scenarios shown (Figure 6), the means for the two groups are the same but their distributions are very different. In the low variability example, the distributions for the two groups are very narrow and overlap only slightly. In the medium variability example the distributions overlap much more, and in the high variability example the two distributions overlap almost entirely. A t-test looks at the ratio of the difference in group means to variability, in essence taking the ratio of the "signal" (the means) versus the "static" (the variance). Click animate below to see the formula.
Animated t-test formula [ animate ]
This ratio is the t-statistic, and the value of this statistic is used to determine the p-value for your test. The t-statistic is referenced to a statistical table and this determines the probability of that result being due to chance alone. Recall that a p-value equal to or less than 5% (0.05) indicates that the two groups are significantly different from one another. Our goal for this course is for you to gain experience with the t-test and to realize its usefulness in making conclusions when comparing groups of data. While we are not stressing the complexities and underlying mechanics of the t-test, you should be able to understand the test, use it to compare data sets, and correctly interpret the results the test gives you. That having been said, let's continue on and do an example to show you the usefulness of the t-test.
The t-test - An Example
Let's assume that a researcher is attempting to determine the effects of pesticide pollution on the hatching success of fish eggs. He has noticed that fish eggs in streams near agricultural fields fare poorly, while those is undisturbed areas hatch successfully. The researcher sets up a lab experiment in which ten groups of fish eggs are allowed to develop in unmanipulated stream water, and ten groups of eggs develop in stream water with the addition of pesticide. The proportion of eggs hatching in each group is tallied (Table 1), and descriptive statistics for the two groups compared.Statistic / No Pesticide / Pesticide
Mean proportion eggs hatching / 0.87 / 0.59
Standard deviation / 0.07 / 0.09
/ Group / Proportion eggs hatching
No Pesticide / Pesticide
1 / 0.80 / 0.50
2 / 0.76 / 0.45
3 / 0.81 / 0.68
4 / 0.90 / 0.77
5 / 0.95 / 0.64
6 / 0.84 / 0.60
7 / 0.88 / 0.54
8 / 0.99 / 0.57
9 / 0.86 / 0.62
10 / 0.93 / 0.57
Table 1. Effects of pesticide pollution on hatching success of fish eggs.
While the results indicate an adverse effect of pesticide exposure on egg hatching success, how can we be sure that the difference in the hatching success is "significant", and didn't just occur by chance? After all, the two data sets do overlap as the lowest value in the no pesticide groups and the highest value in the pesticide groups was 76% hatching success. To compare these two data sets, we must first state our hypothesis. Our null hypothesis would be that the two groups are not different, and that both data sets belong to the same distribution.
Ho:The mean proportion eggs hatching in the two groups is not different
OR
Ho:The two experimental groups are part of the same distribution
We then conduct a t-test on the data set, which will examine the differences in mean and dispersion in the two groups, and provide a probability that the two groups are part of the same distribution.
As the p-value returned by the test was less than 0.05, we can reject our null hypothesis, conclude that the two groups are indeed from different distributions, and that they are significantly different from one another. The researcher can therefore conclude that pesticide exposure reduces the hatching success of eggs of this species. All of the comparisons you will be making in laboratory exercises this semester will mimic this example, so you should take special care to ensure you understand the operation and usefulness the t-test for comparing data sets. / Comparison / t-statistic / p-valuet-test / 7.56 / p< 0.0005
Performing t-tests
To perform t-tests on data sets, we suggest using an online t-test calculator from Graphpad.com. It can be accessed at the address below, and is exceptionally easy to use. WebCrunch, the program we are using to calculate descriptive statistics and create graphs, also has a t-test function, but we suggest you use the one below as it is tailored to the needs of a non-majors science course.
Graph Pad Statistical Application
GraphPad Software Inc.
(1.) Select the "Continuous data" option, then the "Continue" button.
(2.) Select the "t test to compare two means" option, then hit the "Continue" button.
(3.) Simply enter your data in the columns by group, select "Unpaired t-test", and then hit the "Calculate now" button. Your p-value and t statistic will be listed on the results page.
t-tests and Statistical Assumptions - A Word of Caution
Those of you familiar with the t-test have likely noticed that we are omitting a step in our use of the t-test - the testing of assumptions. For a given data set to be suitable for analysis with a t-test, it must meet two assumptions: (1.) the variance in the two groups being compared cannot be significantly different from one another, and (2.) the data must roughly fit a normal distribution. When statisticians and scientists conduct a t-test, they first verify these assumptions with statistical tests, and only proceed once these assumptions have been satisfied. If the variances in the two groups differ appreciably, the data can be mathematically "transformed" to bring variances closer together. If the data are not normally distributed, they can be transformed for normality, or an alternative test that does not require a normal distribution can be used.
As these steps appreciably increase the statistical complexity of t-test analysis, we will not be testing data sets for assumptions in this course. You must therefore realize that the statistical rigor of your results may not be comparable to that in published scientific studies, and that we are consciously avoiding the use of assumption tests to simplify the statistical analyses used in this exercise.
Answers to Practice Problems
PRACTICE PROBLEM #1
p = 0.11Reject or Do Not Reject Ho?
p = 0.56Reject or Do Not Reject Ho?
p = 0.99Reject or Do Not Reject Ho?
p = 0.01Reject or Do Not Reject Ho?
p < 0.005Reject or Do Not Reject Ho?
Sample Problems
The link below will take you to a PDF file with sample problems. Complete the problems assigned by your instructor. If none were assigned, complete problems #1-4.
Sample Problems
Creating and Analyzing Experiments: Putting It All Together
The activities in this module have showed you how to critically evaluate experimental design, calculate and evaluate descriptive statistics for data sets, and compare data sets with inferential statistics. We will now put all of this material together in a “Capstone” exercise. You will design an experiment gather data, calculate descriptive statistics, and calculate and interpret a t-test to compare your groups.
The Metric System
If you are measuring something, you need "units" to describe the object. In formal terms, a scale of measurement is the assignment of numbers or symbols to measure an attribute. In the past, natural units of measurement, such as a "foot", were commonly used. Unfortunately, these units were somewhat arbitrary. In Roman times, for example, a "foot" in England was 29.6 centimeters. When the Saxons took over, the size of a "foot" grew to 33.5 cm. Five centuries later, it was reduced to 30.5 cm. Finally, in 1959, the "International Foot" was defined as 30.48 cm. Even today, a "foot" in England is different from a "fod" in Denmark (31.41 cm), a "fod" in Sweden (29.69 cm), and a "fuss" in Germany (31.61 cm). With the increase in international trade during the 18th century, merchants needed to standardize units of measurement. This resulted in the development and nearly universal adoption of the metric system around the world. Of course, the United States is a notable exception to this worldwide trend, as we continue to use the English system of measurement. We buy our gas in gallons, measure our weight in pounds, and gauge driving distances in miles. The metric system has crept into our society somewhat (e.g., the two-liter soda bottle), but universal acceptance of this system of measurement anytime soon is unlikely.